Special Issue Preface pubs.acs.org/JPCB
Cite This: J. Phys. Chem. B 2018, 122, 3206−3209
Scientific Autobiography of Benjamin Widom
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EARLY YEARS I was born in 1927 in Newark, NJ, where I spent my early grade-school years. We moved to Detroit, MI, in 1937 but remained for only two years, so only long enough for me to have finished the first term of junior high school, when we then moved back east, this time to Brooklyn, NY, first to the Brownsville section and then to East New York. That was to remain my home until I left for graduate school in 1949. On finishing junior high school in Brooklyn, I took and passed the entrance exam for Stuyvesant High School, which I then attended (commuting to Manhattan by subway) until my graduation in midyear of 1944−45. Those were still the war years, when the schools were on an accelerated schedule. Stuyvesant was, and still is, an outstanding school with the highest standards in all subjects but in particular in the sciences and mathematics. I had already, in junior high school, become interested in chemistry, as usual with a home chemistry set, and that interest was furthered at Stuyvesant. I had the opportunity, after having had the introductory chemistry course, of taking an elective in qualitative analysis, which was most interesting. There were also advanced courses in mathematics, which I enjoyed and which later would prove useful. On graduating from high school, I first attended Brooklyn College, which was tuition-free for residents of New York, but after one term there, I was awarded a Pulitzer Scholarship to Columbia, which would then also be tuition-free, so affordable, and I chose to transfer. Brooklyn College, I know, also had the highest standards and would have provided an excellent education, but Columbia seemed an opportunity I should not pass up. I then became again a subway commuter to Manhattan. I was at Columbia for one term when, since I was 18, I received in 1945 my draft notice for induction into the Army. The war was then in its last stages, and by the time I was actually inducted, in March 1946, it had already ended. After one year, in March 1947, the U.S. Army decided it no longer needed my services, I was honorably discharged, and returned to Columbia, this time with the G.I. Bill as well as the continuation of the Pulitzer Scholarship for support. My major was Chemistry. Because I had “missed” a year, 1945−46, I felt the need to accelerate at Columbia and took full loads of summer courses in the summers of 1947 and 1948, which at that time it was still possible to do, and I was then ready to graduate in February 1949. Perhaps because of my interest in mathematics from my high-school days, I had found the mathematical content of physical chemistry, particularly in its theoretical aspects, especially appealing and resolved to pursue it in graduate school. The professor in my physical chemistry course at Columbia was Louis P. Hammett, a famous chemical kineticist and one of the pioneers of physical organic chemistry. His name is still attached to a class of linear free-energy relationships. At that time, and perhaps still, few graduate chemistry programs were willing to admit students in the middle of an © 2018 Benjamin Widom
academic year, yet I did not want to have to wait another half year to start graduate work. Of the programs to which I had applied, only two, Cornell and Berkeley, were willing to have me start in February. Of the two, I preferred Cornell because of its (relative) proximity to New York City, and Professor Hammett assured me their faculty were outstanding. Thus it was that one morning, in February 1949, I left Penn Station in Manhattan at 11:05 on the Black Diamond (since extinct) of the Lehigh Valley Railroad and arrived in Ithaca, NY, at 6:49 that evening. I dragged a heavy suitcase, filled mostly with books, through knee-high snow up the long, steep hill from the railroad station in the valley to my assigned dormitory room in the temporary wooden barracks that had originally been built to house the participants in the Army Specialized Training Program during the war, located on Cornell’s West Campus overlooking Cayuga Lake. The next morning I was in the Baker Laboratory of Chemistry.
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CORNELL, 1949−52 On arriving, I expressed my wish to do my Ph.D. research under the direction of John K. Bragg, but by the fall term of that year, he had already left Cornell for a position at the General Electric research center in Schenectady, NY (which is where he and Bruno Zimm later wrote their famous paper on a model of the helix−coil transition in biopolymers). I was fortunate that Simon Bauer then offered to take me on to do a theoretical project on energy exchange in molecular collisions in connection with his experimental program in chemical kinetics. In the meantime, I was busy taking courses, more in physics and mathematics than in chemistry. I had graduate courses in theoretical mechanics and electrodynamics from Philip Morrison, introductory quantum mechanics and then statistical mechanics from Hans Bethe, and mathematical methods in physics and differential equations from Mark Kac. My examination and oversight committee consisted of Simon Bauer (chairman), Hans Bethe, and Mark Kac. My research project with Simon was “Exchange of Energy in Molecular Collisions” with emphasis on vibrational deexcitation, that is, the conversion of vibrational excitation in one molecule into relative translational energy of the collision partners. I did this in a semiclassical version of the quantum mechanical distorted-wave approximation, within which, inspired by the Landau−Zener theory of nonadiabatic transitions, I treated the relative translational motion classically. Our main application was to the de-excitation of vibrationally excited carbon dioxide on collision with water molecules, for which there was much experimental data. We submitted an abbreviated account of this work to the Journal of Chemical Physics. It was my first publication. I had by then passed my oral examinations, my thesis had been accepted, and I had fulfilled all of the requirements for the Special Issue: Benjamin Widom Festschrift Published: April 5, 2018 3206
DOI: 10.1021/acs.jpcb.8b00128 J. Phys. Chem. B 2018, 122, 3206−3209
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obtain only a lower bound on the radius of convergence, too low a bound to answer the question, but I conjectured that the radius of convergence was infinite and that there is no connection between the phase transition and a divergence of the virial series. My paper later attracted the attention of mathematicians. The definitive result was obtained by Wolfgang Fuchs in the Mathematics Department at Cornell, whose estimate bracketed the radius of convergence between two finite numbers the smaller of which lay above the known transition. Thus, my guess that the radius of convergence was infinite was wrong, but that the possible divergence of the series would be unrelated to the transition was right.
Ph.D. That was 1952 (the Ph.D. was formally granted in 1953), and I was thinking of postdoctoral work. I knew of O. K. Rice in the Chemistry Department of the University of North Carolina from his own work on the quantum mechanics of inelastic molecular collisions. I applied to him for a postdoctoral position and was accepted.
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UNIVERSITY OF NORTH CAROLINA, 1952−54 By the time I arrived in Chapel Hill, Oscar Rice’s interests had largely shifted to phase transitions and critical phenomena, on both the experimental and theoretical sides, and he invited me to collaborate on the theory. His interests included superfluid helium, and while I was still learning about critical phenomena in ordinary fluids and in liquid mixtures, we wrote a paper together on the thermodynamics of the thick film that superfluid helium forms on solid surfaces. That was then my second paper, the first from my time as a postdoctoral. In the meantime, I had learned from Oscar about the paradox of critical-point exponents: that all the equations of state then in use, starting with that of van der Waals, yielded liquid−vapor coexistence curves that were parabolic in the neighborhood of the critical point, while experiment, as was already suspected in van der Waals’s day and was seen most convincingly in Guggenheim’s 1945 corresponding-states plot, showed them universally to be more nearly cubic. This implied an unanticipated singularity in the thermodynamics of the critical point. It was reinforced when Bruno Zimm, in a 1952 paper, showed that Rice’s earlier experimental measurements of the coexistence curve near the critical solution point of a binary liquid mixture, and his own data on another such mixture, when appropriately scaled, could be superimposed on the liquid− vapor coexistence curves of pure fluids, so again with that universal, near-cubic shape. Rice recognized that such a singularity in the coexistence curve at the critical point must be accompanied by a corresponding one in the critical isotherm. He argued that its algebraic degree would be greater by 1 than that in the coexistence curve, so close to 4. We analyzed the available experimental data and found that degree to be indeed close to 4, perhaps a little greater. (Some years later, it would be recognized that the degree of the critical isotherm exceeds that of the coexistence curve by more than 1 and that the former is really closer to 5 than to 4, but as of 1954, the data were not good enough to show that. It is now best seen in the thermodynamically analogous paramagnetic-to-ferromagnetic transition near its Curie point.) That was the only other paper of which Oscar and I would be coauthors, but I did additional work in which he took an interest and which he encouraged. It resulted in two further publications. One was on the distribution of the zeros and poles of the configurational partition function in the complex volume plane, which was inspired by the 1952 Yang−Lee paper on the distribution of the zeros of the grand partition function of the two-dimensional lattice gas in the complex fugacity (activity) plane, and its connection to the phase transition and critical point. The second was on the convergence of the virial series of the ideal Bose−Einstein gas. At that time, there was much speculation about the possibility that the virial series for an ordinary classical fluid may have a finite radius of convergence and that its divergence would mark the liquid−vapor phase transition. My thought was to test that idea in the ideal Bose− Einstein gas, for which the location of the onset of the Einstein condensation was known analytically. In the end, I was able to
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CORNELL, 1954− My two-year appointment as Oscar Rice’s postdoctoral associate was coming to an end, and it was time for me to find another job. I received two offers of an instructorship in a Chemistry department, from Cornell and from Harvard. They offered identical salaries (collusion?), a little less than my postdoctoral salary from Oscar’s ONR grant. At the time, the Cornell department required three years of postdoctoral work before an appointment as assistant professor, and I had only two, so their offer was as an instructor but with the promise of promotion to assistant professor after the first year. Harvard made no such promise. Besides, I, and my wife Joanne, a Cornell graduate, were both delighted with the chance of returning, so there was no contest. The remainder of this account will be an abbreviated version of an earlier scientific autobiography I had the occasion to write (2011). This will no longer be chronological but arranged by research area for coherence. I give the dates or the ranges of dates of the work referred to. The individual publications can then be readily located in the appended list. I also emphasize earlier over later work just for the chance to reminisce.
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CRITICAL PHENOMENA While still working with Oscar Rice at UNC, I had tried hard to invent a simple equation of state that would incorporate the known nonclassical critical-point exponents, but I did not come up with anything very convincing. Then, in 1963, while Edward Guggenheim was in our Chemistry Department at Cornell for a few weeks as our Baker Lecturer, he invited me to spend a term in his department in Reading, in the U.K. My visit was finally arranged for the spring term of 1965. While there, and giving a series of lectures, I was still thinking about the equation-of-state problem. At one point, I tried one that (after some hard thermodynamics!) yielded a constant-volume heat capacity that diverged logarithmically at the critical point. That astonished me because Onsager in 1944 had famously found such a divergence of the heat capacity in the two-dimensional Ising model at its critical point and something much like it was seen experimentally in the analogous order−disorder transition in beta-brass. When I looked to see what features of my equation of state had led to it, I found it was a certain homogeneity in its structure as a function of the distances of the temperature and density from their critical-point values. That generalized form of the relation (1965) led to verifiable relations among critical-point exponents and has since become an accepted representation of the thermodynamics at critical points. 3207
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equation, but with the hard-sphere term not that of spheres in one dimension, which it is in the vdW equation itself, but that of spheres in three dimensions, as had been determined in earlier computer simulations by Alder and Wainwright. The result was then “a rigid sphere model for the melting of argon” (1964), in which the agreement with experiment was good enough to be convincing. In a different contribution to the theory of liquids, Michael Fisher and I (1969) considered the mode of decay of the paircorrelation function at long distances, which could be either as a monotonically decaying exponential or oscillatory with an exponentially decaying amplitude. We analyzed exactly soluble one-dimensional models and found loci in their pressure− temperature and density−temperature planes across which the decay changed between monotonic and oscillatory: at any temperature, oscillatory at the high pressures and densities, monotonic at the lower. We then conjectured what such loci might be in real fluids, where it was already known to be monotonic near critical points, and supposed generally (1967) to be oscillatory when the repulsive forces between molecules were dominant and monotonic when the attractive forces were.
In a companion paper (1965), I proposed a relation between the vanishing of the surface tension at the critical point and the divergence of the correlation length; i.e., of the length that characterizes the exponential decay of the pair-correlation function at large separations. This then related two additional critical-point exponents to each other and to those present in the equation of state. I also found an alternative version of this relation in which the system’s spatial dimension appears explicitly. This was the first of a class of explicitly dimensionality-dependent critical-point exponent relations obtained by combining it with those already known, which held independently of the spatial dimension. That any individual exponent would depend on the spatial dimension had already been clear from the early work of Onsager on the two-dimensional Ising model, and from that of Domb, Fisher, and others in Cyril Domb’s group in King’s College, London, on the two- and three-dimensional models. What was new was the explicit appearance of the dimensionality as a (continuous!) variable in formulas relating the exponents to each other. For the spring term of 1969 (coincidentally a difficult time for Cornell, as it was for many universities), I was on leave of absence at Imperial College in London. I was mostly visiting John Rowlinson. It was there that we worked on what we came to call the “penetrable-sphere” model (1970). (He has a paper about it in this issue.) It was a model that starts as a mixture of two species of molecules where like molecules do not interact with one another but unlike molecules repel as hard spheres. Then, on integrating out the degrees of freedom of one of the two components, we have an equivalent one-component model of interacting, penetrable spheres, hence the name. Many of its properties can be obtained analytically. It has a range of twophase coexistence ending in a critical point, and it provided an early example of the violation of the “law” of rectilinear diameters. Critical phenomena (later including critical end points and tricritical points) and interfacial structure and tension (including wetting and prewetting transitions and line tension) continued to be among the subjects of my study during the whole of my time at Cornell, even to the present. We were also active on the experimental side during much of this time (1960, 1973−76, 1980−88, 1992)that is to say, while I was mostly an interested spectator, my talented students and associates did the experiments (with the exception of no. 49 on the publication listmy first and last personal foray into experiment).
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CHEMICAL KINETICS
On returning to Cornell from my time as a postdoctoral, I had not forgotten my earlier graduate work in connection with Simon Bauer’s program in energy exchange in molecular collisions and its role in reaction kinetics. I almost immediately (1955) resumed thinking about those matters along with my other work (and even made a foray into that area as “recently” as 1998, treating a model of molecular motors). Just as in my work on the statistical mechanics of liquids and on phase equilibria and critical phenomena, I often here, too, considered analytically tractable models in which, if lucky, one might discern general principles. Among those I studied was a class of stochastic models (1965, 1971, 1974) in which, in terms of the eigenvalues of the matrix, or of the integral kernel, of the postulated microscopic transition probabilities, one could derive the phenomenological forward and reverse rate constants of an associated chemical reaction. Under certain conditions on the transition probabilities, the eigenvalues were found to satisfy a Schrödinger equation (without Planck’s constant!) for a particle moving in a two- (or more-) well potential, with the wells in near resonance and with high barriers between them. Each well corresponded to a chemical species, and the forward and reverse rate constants for their interconversion were related to the small tunneling splitting between the low-lying eigenvalues. Such models continue to find application in the analysis of the kinetics of the simultaneous reactions among several identifiable species in a reaction mixture.
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THEORY OF LIQUIDS This was mostly on the equilibrium statistical mechanics of liquids. In 1962−63, I found a relation that expressed the chemical potentials in a one- or multi-component liquid in terms of the distribution of the potential energy felt by an inserted “test” or “ghost” particle. Likewise, the insertion of a ghost pair led to an expression for the pair-correlation function. The relations have variously been called the potential-distribution theory or the particle-insertion theory or method. Its main use soon came to be in the calculation of chemical potentials in computer simulation, although my initial applications of the method were in obtaining approximate equations of state or some properties of pair-distribution functions analytically. Christopher LonguetHiggins and I made an application of the principle to derive, from a physical picture of a simple liquid as consisting of attracting hard spheres, an analogue of the van der Waals
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BOOKS In 1978, I was in Oxford for a term giving a series of lectures and again spent some time with John Rowlinson, who had in the meantime moved there from Imperial College. As my time there was coming to an end, he suggested we write a book about theories of interfaces. I immediately agreed. With John as the driving force of our collaboration, it became the “Molecular Theory of Capillarity”. It was published in 1982. A translation into Russian by A. I. Rusanov and V. L. Kuzmin appeared in 1986. 3208
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Sometime in the 1990s, there was a plan for a multi-volume, multi-author undergraduate physical chemistry textbook, to include a brief treatment, no more than 10% of the whole, on statistical mechanics. My colleague Paul Houston, who had agreed to do the chemical kinetics and reaction dynamics volume, recruited me to do the statistical mechanics. In the end, only Paul and I completed our parts, so the project had to be abandoned. Paul then published his as a separate volume with a different publisher. I did nothing with mine until, sometime around 2000−01, I had an invitation from Cambridge University Press to write a book on statistical mechanics! I told them what I already had and what its history was. They asked to see it and ultimately published it as “Statistical Mechanics: A Concise Introduction for Chemists” (2002). It has proved useful. My long-time collaborator, Professor Kenichiro Koga, of Okayama University, translated it into Japanese; his translation appeared in 2005.
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ACKNOWLEDGMENTS I wish to express my gratitude to my colleague Roger Loring and my research collaborators Dor Ben-Amotz and Kenichiro Koga for their initiative and their heroic efforts in bringing this special issue into existence. Of course, much of the work represented by the publications in the appended list (and related ones of which I was nominal supervisor but not coauthor) would not have been possible without the dedication and expertise of the many co-workers I have been privileged to have over the years. It is a pleasure for me to have this opportunity to acknowledge their contributions.
Benjamin Widom Department of Chemistry, Cornell University
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DOI: 10.1021/acs.jpcb.8b00128 J. Phys. Chem. B 2018, 122, 3206−3209
